Upload
others
View
1
Download
0
Embed Size (px)
Citation preview
The Effect of Estimation in High–dimensional Portfolios
Luitgard A. M. Veraart
Joint work with Axel Gandy, Imperial College London
Analysis, Stochastics, and ApplicationsVienna University, July 2010
Luitgard A. M. Veraart (KIT) Estimation in High–dimensional Portfolios June 2010 1 / 12
Outline
1 Classical Portfolio Optimisation
2 Plug-In Strategies with Estimated Parameters
3 James-Stein-Shrinkage Applied to Strategies
4 L1–Constrained Strategies - LASSO
5 Other Strategies
6 Application to Empirical Data
Luitgard A. M. Veraart (KIT) Estimation in High–dimensional Portfolios June 2010 2 / 12
Optimal Portfolio SelectionThe asset prices: 1 bond S0(t) = ert , d risky assets
dSi (t) = Si (t)[µidt +d∑
j=1
σijdWj(t)], Si (0) > 0, i = 1, . . . , d ,
r > 0 interest rate, µ ∈ Rd drift, σ ∈ Rd×d volatility matrix of fullrank (all constant), W d-variate Brownian motion.Investor has T > 0 fixed time horizon, X0 > 0 constant initial wealth,chooses πi (t) fraction of the wealth invested in the ith asset at time
t, resulting in time-t-wealth Xt with dXt =∑d
i=0 πi (t)XtdSi (t)Si (t) .
Investor seeks π to maximise V (π) := E[log(XT )].Optimal solution
π∗ = Σ−1(µ−r1),
where π0(t) = 1−∑d
i=1 πi (t), π = (π1, . . . , πd)T , Σ = σσT .
The Problem
What if we need to estimate µ?What if the number of risky assets d →∞?
Luitgard A. M. Veraart (KIT) Estimation in High–dimensional Portfolios June 2010 3 / 12
Optimal Portfolio SelectionThe asset prices: 1 bond S0(t) = ert , d risky assets
dSi (t) = Si (t)[µidt +d∑
j=1
σijdWj(t)], Si (0) > 0, i = 1, . . . , d ,
r > 0 interest rate, µ ∈ Rd drift, σ ∈ Rd×d volatility matrix of fullrank (all constant), W d-variate Brownian motion.Investor has T > 0 fixed time horizon, X0 > 0 constant initial wealth,chooses πi (t) fraction of the wealth invested in the ith asset at time
t, resulting in time-t-wealth Xt with dXt =∑d
i=0 πi (t)XtdSi (t)Si (t) .
Investor seeks π to maximise V (π) := E[log(XT )].Optimal solution
π∗ = Σ−1(µ−r1),
where π0(t) = 1−∑d
i=1 πi (t), π = (π1, . . . , πd)T , Σ = σσT .
The Problem
What if we need to estimate µ?What if the number of risky assets d →∞?
Luitgard A. M. Veraart (KIT) Estimation in High–dimensional Portfolios June 2010 3 / 12
Plug-in Merton Strategy with Estimated µ
General Unbiased Plug-in Estimator
Estimate µ by µ and define the plug-in strategy π = Σ−1(µ−r1).
Assume π ∼ N(Σ−1(µ− r1),V 20 ), V0 ∈ Rd×d , then
V (π) = V (π∗)− T
2trace(ΣV 2
0 ).
Specific Plug-in Estimator
Observation period [−test , 0] for test > 0.
Set
µi =log(Si (0))− log(Si (−test))
test+
1
2
d∑j=1
σ2ij .
Then π ∼ N(Σ−1(µ− r1),Σ−1/test).
V (π) = V (π∗)−d T2test
.
There are realistic scenarios in which even V (π)→ −∞ as d →∞.
Luitgard A. M. Veraart (KIT) Estimation in High–dimensional Portfolios June 2010 4 / 12
James-Stein-Type Shrinkage of the Strategy
The James-Stein-Strategy
Let π = Σ−1(µ− r1), π0 ∈ Rd , a > 0 fixed constants. Consider
πJS ,π0
=
(1− a
(π − π0)TΣ(π − π0)
)(π − π0) + π0.
The Expected Utility for the JS-Strategy
Let µ ∼ N(µ,Σ/test), K ∼ Poisson(λ), λ = (π∗ − π0)TΣ(π∗ − π0)/2:
V (πJS ,π0) = V (π) +
T
2a
[2d − 2
test− a
]E[
testd − 2 + 2K
].
πJS ,π0
dominates π for 0 < a < 2(d − 2)/test ; optimal a = (d − 2)/test .
Special Choices for π0 and Optimal a
π0 = π∗: V (πJS ,π0) = V (π∗)− T
test.
π0 = βd 1, β ∈ R: In some situations V (πJS ,π
0)→∞ as d →∞.
Luitgard A. M. Veraart (KIT) Estimation in High–dimensional Portfolios June 2010 5 / 12
L1–constrained Strategies - LASSO
General Idea
Require that π satisfies ‖π‖1 =∑d
i=1 |πi | ≤ c for a constant c ≥ 0.
V (π) ≥ log(X0) + rT − TE{c maxi |µi − r |+ c2
2 maxi ,j |Σij |}
.
If maxi |µi − r |, maxi ,j |Σij | bounded, V (π) 6→ −∞ as d →∞.
Specific Results
For Σ = η2(ρ11T + (1− ρ)I ), η > 0, 0 ≤ ρ ≤ 1 analytic results for
the optimal L1-constrained strategies, if µ known.
for the L1-constrained plug-in strategy as d →∞:I the distribution of #{i : π∗i 6= 0}, if ρ = 0,I an upper bound on limd→∞ P(#{i : π∗i 6= 0} > k), if ρ > 0.
Luitgard A. M. Veraart (KIT) Estimation in High–dimensional Portfolios June 2010 6 / 12
Other Strategies and Performance for d →∞
Other Norm Constraints
L0-restricted strategies: no degeneration of expected utility asd →∞.
L2-restricted strategies: degeneration possible.
Special L1-Constraints
1/d-strategy: Strategy that invests the same amount into all stocks,i.e. πc/d = c
d 1 for some c > 0.
Equal Weighting of the most Extreme stocks (EWE):
πEWEki
=c
βdsign(aki )I(i ≤ βd), i = 1, . . . , d ,
where ai = µi−rΣii
, c > 0, β ∈ (0, 1) constants, ki are such that|ak1 | > |ak2 | > · · · > |akd |.
Luitgard A. M. Veraart (KIT) Estimation in High–dimensional Portfolios June 2010 7 / 12
Example - Trading S&P500
Stocks in S&P 500 index on 01/01/2006 having daily returns for alltrading days between 2001 and 2008 (373 stocks, n=2011 tradingdays, daily returns).
Specific random ordering of stocks. Allow the strategies to invest inthe first d stocks of this ordering.
X0 = 1, r = 0.02, roughly T = 1.
Use of unbiased estimators based on observed stock prices at timepoints 0,∆, 2∆, . . . , (n − 1)∆:
µdata =1
∆ξ +
1
2diag(Σdata),
Σdataµ,ν =
1
∆(n − 2)
n−2∑i=0
[Rµ(i)− ξµ
] [Rν(i)− ξν
]for µ, ν = 1, . . . , d , where Rµ(i) = log
(Sµ((i+1)∆)
Sµ(i∆)
),
ξµ = 1n−1
∑n−2i=0 Rµ(i).
Luitgard A. M. Veraart (KIT) Estimation in High–dimensional Portfolios June 2010 8 / 12
Analytic and Simulation Results−2
0−1
00
1020
d
n == 504
expe
cted u
tility
V((ππ|µµ))
Merton−Strategies
µµ,ΣΣ knownµµ estimated, ΣΣ knownµµ,ΣΣ estimated
d
n == 1008
d
n == 2016−2
0−1
00
1020
d
expe
cted u
tility
V((ππ|µµ))
James−Stein Strategies
JS, ππ0 == ππ*JS, ππ0 == ππ*, ΣΣ est.JS, ππ0 == 0JS, ππ0 == 0, ΣΣ est.
d d
0 100 200 300
0.00.1
0.20.3
0.40.5
dd
expe
cted u
tility
V((ππ|µµ))
L1−restricted Strategies
L1L1,µµ est.L1,µµ,ΣΣ est1 dEWE, ββ == 0.1
0 100 200 300dd
0 100 200 300dd
Expected utility plotted against the number d of available stocks.
Out-of-Sample Performance2005
−1
−0.5
0
0.5
1
log((
XT))
0 100 200 300
−− ∞∞
−20
−10
0
10
log((
XT))
d
2006
0 100 200 300
d
2007
0 100 200 300
d
2008Strategy
1/dL1EWE, ββ == 0.1
0 100 200 300
Strategy
MertonJS
d
log(XT ) with T = 1 year plotted against the number d of available stocks.
Luitgard A. M. Veraart (KIT) Estimation in High–dimensional Portfolios June 2010 10 / 12
Summary
Main Contributions
Quantification of the effect of estimation in vast portfolios(unknown µ and large d).
Analysis of strategies which are less affected by estimation.
Analytic formulae for James-Stein and optimal L1-constrainedstrategies.
Specific Conclusions
Estimation effects must not be ignored in vast portfolios!
Simple plug in strategies have a loss through estimation linear in d .
James-Stein shrinkage performs better than simple plug in strategies.
L1-constrained strategies cannot degenerate.
L1-constrained strategies and particularly the EWE-strategy and 1/dstrategy perform well also in out-of-sample tests.
Luitgard A. M. Veraart (KIT) Estimation in High–dimensional Portfolios June 2010 11 / 12
References
Brodie, J., Daubechies, I., De Mol, C., Giannone, D. & Loris, I.(2009). Sparse and stable Markowitz portfolios. P. Natl. Acad. Sci.106, 12267–12272.
Fan, J., Zhang, J. & Yu, K. (2009). Asset allocation and riskassessment with gross exposure constraints on vast portfolios.Preprint, Princeton University.
James, W. & Stein, C. (1961). Estimation with quadratic loss. InProc. Fourth Berkeley Symp. on Math. Statist. and Prob., Vol. 1(Univ. of Calif. Press), 361–379.
Merton, R. (1971). Optimum consumption and portfolio rules in acontinuous-time model. J. Econ. Theory 3, 373–413.
Stein, C. M. (1981). Estimation of the mean of a multivariate normaldistribution. Ann. Statist. 9, 1135–1151.
Tibshirani, R. (1996). Regression shrinkage and selection via the lasso.J. Roy. Stat. Soc. B 58, 267–288.
Luitgard A. M. Veraart (KIT) Estimation in High–dimensional Portfolios June 2010 12 / 12
Analytic Results for LASSO with Σ = η2I
Optimal L1-constrained strategy with known µ
Suppose |µ1 − r | > |µ2 − r | > . . . > |µd − r |. Then π† = 1η2 (µ− r1) is a
solution to the L1-constrained optimisation problem if ‖π†‖1 ≤ c .Otherwise, the unique solution is
π∗:=1
η2(sign(µ1 − r)(|µ1 − r |−a), . . . , sign(µk − r)(|µk − r |−a), 0, . . . , 0)T,
k = min{l ∈ {1, . . . , d} : c ≤ 1
η2
∑li=1(|µi − r | − |µl+1 − r |)
},
a = 1k
[η2c −
∑ki=1 |µi − r |
]and µd+1 = r .
Plug-in strategy with iid normally distributed estimators µ1, . . . , µd
Let c = αcd , cd > 0 norming constants from extreme value theory forfolded normal distribution FN(E(µ1 − r),Var(µ1)). Then
#{i : π∗i 6= 0} L→ K + 1 (d →∞),
where K is a Poisson distribution with expected value αη2.Luitgard A. M. Veraart (KIT) Estimation in High–dimensional Portfolios June 2010 13 / 12